Process Cost Analysis System

ABSTRACT

A method of process cost analysis that includes determining a per-unit cost function for executing a process step, determining a percentage-of-acceptable-parts function for executing a process step, and receiving production data into memory. The production data corresponds to a measured quality metric of the executed process step. The method further includes determining a probability density function for the received production data, executing on a processor a correlation routine for cross-correlating the cost function with the probability density function of the production data to provide a first cross-correlation, and executing on the processor the correlation routine for cross-correlating the percentage-of-acceptable-parts function with the probability density function of the production data to provide a second cross-correlation. The method includes determining an average effective per-unit cost to produce a resultant of the process step by dividing the first cross-correlation by the second cross-correlation.

CROSS REFERENCE TO RELATED APPLICATIONS

This U.S. patent application claims priority under 35 U.S.C. §119(e) toU.S. Provisional Application 61/433,442, filed on Jan. 17, 2011, whichis hereby incorporated by reference in its entirety.

TECHNICAL FIELD

This disclosure relates to process cost analysis and statistical processcontrol systems.

BACKGROUND

Statistical process control (SPC) typically entails the application ofstatistical methods to monitor and control a process to ensure that theprocess operates at its full potential to produce conforming product.Under SPC, a process may behave predictably to produce as muchconforming product as possible with the least possible waste. While SPChas been applied most frequently to controlling manufacturing lines, itapplies equally well to any process with a measurable output.

In general, SPC can be used to examine a process and the sources ofvariation in that process using tools that give weight to objectiveanalysis over subjective opinions and to determine the strength of eachsource numerically. Variations in the process that may affect thequality of the end product or service can be detected and corrected,thus reducing waste as well as the likelihood that problems will bepassed on to a customer. With an emphasis on early detection andprevention of problems, SPC generally has a distinct advantage overother quality methods, such as inspection, that apply resources todetecting and correcting problems after they have occurred.

SUMMARY

One aspect of the disclosure provides a method of process cost analysis.The method includes determining a per-unit cost function for executing aprocess step, determining a percentage-of-acceptable-parts function forexecuting a process step, and receiving production data into memory. Theproduction data corresponds to a measured quality metric of the executedprocess step. The method further includes determining a probabilitydensity function for the received production data, executing on aprocessor a correlation routine for cross-correlating the per-unit costfunction with the probability density function of the production data toprovide a first cross-correlation, and executing on the processor thecorrelation routine for cross-correlating thepercentage-of-acceptable-parts function with the probability densityfunction of the production data to provide a second cross-correlation.The method includes determining an average effective per-unit cost toproduce a resultant of the process step by dividing the firstcross-correlation by the second cross-correlation.

In another aspect, a method of process cost analysis includes receivingat least one cost factor into memory for executing a process step andreceiving production data into memory.

The production data corresponds to a measured quality metric of theexecuted process step. The method also includes determining a per unitmanufacturing cost for the process step, determining a percentage ofacceptable units manufactured by the process step, modeling the per unitmanufacturing cost as a piecewise linear function, and modeling thepercentage of acceptable units as a piecewise constant function. Themethod further includes determining a probability density function forthe received production data, determining a first cross-correlation byexecuting on a processor a correlation routine for cross-correlating thepiecewise linear function with the probability density function for thereceived production data, and determining a second cross-correlation byexecuting on the process the cross-correlation routine forcross-correlating the piecewise constant function with the probabilitydensity function for the received production data. Moreover, the methodincludes determining an average effective per-unit cost to produce aresultant of the process step by dividing the first cross-correlation bythe second cross-correlation.

Yet another aspect of the disclosure provides a method of real-timestatistical process control. The method includes determining a costfunction for executing a process step and receiving production data intomemory. The production data corresponds to a measured quality metric ofthe executed process step. The method further includes determining aprobability density function for the received production data andexecuting on a processor a correlation routine for cross-correlating thecost function with the probability density function of the productiondata. The method also includes determining a percentage of acceptableprocess resultants produced by the process step and determining anaverage effective per-unit cost to produce a resultant of the processstep by dividing the cross-correlation of the cost function and theprobability density function of the production data by thecross-correlation of the percentage of acceptable process resultants andthe probability density function of the production data.

Implementations of the disclosure may include one or more of thefollowing features. In some implementations, the method includesdetermining a minimum value of the quotient of the first and secondcross-correlations to determine a minimum average effective per-unitcost associated with a current data distribution. In someimplementations, the method includes electronically displaying a plotcorresponding to the average effective per-unit cost and/or minimumaverage effective per-unit cost. The method may include determining anoptimal process mean associated with the minimum average effectiveper-unit cost associated with a current data distribution. The methodmay include presenting a process offset based at least in part on adifference between the optimal process mean and a current process meanassociated with a current data distribution. The process offset can beassociated with a cost savings of the process step. The minimum averageeffective per-unit cost can be iteratively determined using differentstandard deviations for the probability density function of the receiveddata corresponding to a theoretical reduction in process variability. Insome examples, the method includes determining an optimal mean andminimum average effective per-unit cost at one or more standarddeviations. The method may include presenting the iteratively determinedminimum average effective per-unit costs. For example, a progression ofminimum average effective per-unit costs may be displayed at various %reductions in variation (e.g., 10%, 20%, 30%, etc.).

In some implementations, the cost function includes at least one of anindirect cost and a direct cost factor associated with the process step.The indirect cost factor may include a process cost factor (e.g., afixed cost of machinery). The direct cost factor may include materialcosts. For example, the direct cost factor may include at least one of alower specification limit violation scrap cost factor, a lowerspecification limit violation rework cost factor, an upper specificationlimit violation scrap cost factor, and an upper specification limitviolation rework cost factor. Additional cost factors may include alower specification limit violation scrap cost factor, a lowerspecification limit violation rework cost factor, an upper specificationlimit violation scrap cost factor, an upper specification limitviolation rework cost factor, a maximum rework attempts cost factor forlower specification limit violations, a maximum rework attempts costfactor for upper specification limit violation, a lower dimensionallimit for rework threshold cost factor, an upper dimensional limit forrework threshold cost factor, a cost of each rework attempt cost factorfor lower specification limit violations a cost of each rework attemptcost factor for upper specification limit violations a percentage chanceof rework success cost factor for lower specification limit violations,a percentage chance of rework success cost factor for upperspecification limit violations, a percentage chance of rework failurecost factor for lower specification limit violations, a percentagechance of rework failure cost factor for upper specification limitviolations, a percentage of lower specification limit violations allowedcost factor a percentage of upper specification limit violations allowedcost factor a disposal cost of a scrapped unit cost factor for lowerspecification limit violations, a disposal cost of a scrapped unit costfactor for upper specification limit violations, a per unit recoveryvalue of a scrapped unit cost factor for lower specification limitviolations, a per unit recovery value of a scrapped unit cost factor forupper specification limit violations, a per unit material value of ascrapped unit cost factor for lower specification limit violations, aper unit material value of a scrapped unit cost factor for upperspecification limit violations, a per step process cost factor, and aper unit of measure material cost factor.

Another aspect provides a method of real-time statistical processcontrol that includes associating indirect cost factors and direct costfactors with at least one process step. The direct cost factor includesat least one of a lower specification limit violation scrap cost factor,a lower specification limit violation rework cost factor, an upperspecification limit violation scrap cost factor, and an upperspecification limit violation rework cost factor. The method furtherincludes receiving production variable data into memory and executing ona processor a first cross-correlation routine for cross-correlating atleast one of the cost factors and the production variable data. Themethod further includes executing on a processor a secondcross-correlation routine for cross-correlating a percentage ofacceptable parts function and a probability density function of theproduction variable data. The method further includes dividing the firstcross-correlation by the second cross-correlation to determine theaverage effective per-unit cost. The method may include determining aminimum average effective per-unit cost associated with a current datadistribution. The method may include iteratively determining the minimumaverage effective per-unit cost using different standard deviations forthe probability density function of the production variable datacorresponding to a theoretical reduction in process variability. In someexamples, the method includes determining an optimal mean and minimumaverage effective per-unit cost at one or more standard deviations.Moreover, the method may include presenting the iteratively determinedminimum average effective per-unit costs.

The method may include executing a selection criteria on the determinedaverage effective per-unit costs and identifying one of the determinedaverage effective per-unit costs as corresponding to a recommendedprocess mean associated with a current data distribution. The processvariable data may allow monitoring of a quality of the at least oneprocess step. When receiving multiple types of process variable data, adirect cost factor can be associated with each type of process variabledata. In some examples, the average effective per-unit cost includes aratio of an actual cost to make an acceptable part to a proportion ofacceptable parts. The method may include electronically displaying aplot corresponding to the determined minimum average effective per-unitcost and/or determining an optimal process mean associated with theminimum average effective per-unit cost associated with a current datadistribution. A process offset may be presented based at least in parton a difference between the optimal process mean and a current processmean associated with a current data distribution. The process offset canbe associated with a cost savings of the process step.

Another aspect of the disclosure provides a cost inspection system for aprocess. The system includes a cost module, a data module, across-correlation module, and a cost module. The cost module receives acost model associated with the at least one process step and provides aper-unit cost function for executing a process step. The cost modulereceives a cost model associated with the at least one process step andprovides a percentage-of-acceptable-parts function for executing aprocess step. The data module receives production data corresponding toa measured quality metric of the executed process step and provides aprobability density function for the received production data. Thecross-correlation module cross-correlates the per-unit cost functionwith the probability density function of the production data to providea first cross-correlation and cross-correlates thepercentage-of-acceptable-parts function with the probability densityfunction of the production data to provide a second cross-correlation.Lastly, the cost module executes on a computing processor and determinesan average effective per-unit cost to produce a resultant of the atleast one process step by dividing the first cross-correlation by thesecond cross-correlation.

In yet another aspect, a real-time statistical process control (SPC)system includes a function module that receives a cost model associatedwith at least one process step and provides a per-unit cost functionassociated with executing the at least one process step and apercentage-of acceptable-parts function associated with executing the atleast one process step. The real-time SPC system includes a data modulethat receives production data corresponding to a measured quality metricof the executed process step and provides a probability density functionfor the received production data. A cross-correlation modulecross-correlates the per-unit cost function with the probability densityfunction of the production data to provide a first cross-correlation andcross-correlating the percentage-of-acceptable-parts function with theprobability density function of the production data to provide a secondcross-correlation. A cost module, executing on a computing processor,determines an average effective per-unit cost to produce a resultant ofthe at least one process step by dividing the first cross-correlation(i.e., the cross-correlation of the cost function and the probabilitydensity function of the production data) by the second cross-correlation(i.e., the cross-correlation of the percentage of acceptable processresultants and the probability density function of the production data).

Implementations of the disclosure may include one or more of thefollowing features. In some implementations, wherein the cost moduledisplays on an electronic device a plot corresponding to the determinedaverage effective per-unit cost and/or determines an optimal processmean associated with the plotted minimum average effective per-unit costassociated with a current data distribution. The cost module may presenta process offset based at least in part on a difference between theoptimal process mean and a current process mean associated with acurrent data distribution. The process offset can be associated with acost savings of the at least one process step. In some examples, thecost module iteratively determines the minimum average effectiveper-unit cost using different standard deviations for the probabilitydensity function of the received data corresponding to a theoreticalreduction in process variability. In some examples, an optimal mean andminimum average effective per-unit cost can be determined at one or morestandard deviations. The cost module electronically may display theiteratively determined minimum average effective per-unit costs.

The cost function may include at least one of an indirect cost factorand direct cost factor associated with the at least one process step.The indirect cost factor can include a process cost factor. The directcost factor comprising at least one of a lower specification limitviolation scrap cost factor, a lower specification limit violationrework cost factor, an upper specification limit violation scrap costfactor, and an upper specification limit violation rework cost factor.Additional cost factors may include a lower specification limitviolation scrap cost factor, a lower specification limit violationrework cost factor, an upper specification limit violation scrap costfactor, an upper specification limit violation rework cost factor, amaximum rework attempts cost factor for lower specification limitviolations, a maximum rework attempts cost factor for upperspecification limit violation, a lower dimensional limit for reworkthreshold cost factor, an upper dimensional limit for rework thresholdcost factor, a cost of each rework attempt cost factor for lowerspecification limit violations a cost of each rework attempt cost factorfor upper specification limit violations a percentage chance of reworksuccess cost factor for lower specification limit violations, apercentage chance of rework success cost factor for upper specificationlimit violations, a percentage chance of rework failure cost factor forlower specification limit violations, a percentage chance of reworkfailure cost factor for upper specification limit violations, apercentage of lower specification limit violations allowed cost factor apercentage of upper specification limit violations allowed cost factor adisposal cost of a scrapped unit cost factor for lower specificationlimit violations, a disposal cost of a scrapped unit cost factor forupper specification limit violations, a per unit recovery value of ascrapped unit cost factor for lower specification limit violations, aper unit recovery value of a scrapped unit cost factor for upperspecification limit violations, a per unit material value of a scrappedunit cost factor for lower specification limit violations, a per unitmaterial value of a scrapped unit cost factor for upper specificationlimit violations, a per step process cost factor, and a per unit ofmeasure material cost factor.

The details of one or more implementations of the disclosure are setforth in the accompanying drawings and the description below. Otheraspects, features, and advantages will be apparent from the descriptionand drawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic view of an exemplary real-time statistical processcontrol (SPC) system.

FIG. 2 is a schematic view of an exemplary real-time SPC system applyinga cost model to a set of variable data to determine an optimal mean ofthe variable data.

FIG. 3 is a schematic view of associating cost factors and variable datato a process step.

FIGS. 4A and 4B are schematic views of associating cost factors andvariable data to multiple process steps.

FIGS. 4C and 4D illustrate an exemplary cost model user interface thatallows a user to create a cost model and define one or more processsteps.

FIG. 5 provides an exemplary arrangement of operations for a method ofreal-time statistical process control.

FIG. 6 provides a graph of cost with respect to a quality metric of anexemplary lathing operation.

FIG. 7A graphically illustrates an exemplary determined cost per unitversus process set point of a process.

FIG. 7B provides a detail view of FIG. 7A.

FIG. 8 is a schematic view of an exemplary administrator window of areal-time SPC system.

FIGS. 9A-9D provide schematic views of various portions of an exemplaryprocess pane.

FIG. 10 is a schematic view of an exemplary material costs pane.

FIG. 11 is a schematic view of an exemplary lower specification limitcosts pane.

FIG. 12 is a schematic view of an exemplary upper specification limitcosts pane.

FIG. 13 is a schematic view of an exemplary administrator view of areal-time SPC system.

FIG. 14 is a schematic view of an exemplary variable analyzer view.

FIG. 15A is a schematic view of an exemplary cost model view receivinginputs for a lower specification limit scrapped violation cost and anupper specification limit scrapped violation cost.

FIG. 15B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 15C provides a detail view of FIG. 15B.

FIG. 16A is a schematic view of an exemplary cost model view receivinginputs for a lower specification limit scrapped violation cost and anupper specification limit rework violation cost.

FIG. 16B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 16C provides a detail view of FIG. 16B.

FIG. 17A is a schematic view of an exemplary cost model view receivinginputs for a lower specification limit rework violation cost and anupper specification limit scrapped violation cost.

FIG. 17B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 17C provides a detail view of FIG. 15B.

FIG. 18A is a schematic view of an exemplary cost model view receivinginputs for a lower specification limit rework violation cost and anupper specification limit rework violation cost.

FIG. 18B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 18C provides a detail view of FIG. 18B.

FIG. 19A is a schematic view of an exemplary cost model view receivinginputs for a material cost and a lower specification limit scrappedviolation cost.

FIG. 19B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 19C provides a detail view of FIG. 19B.

FIG. 20A is a schematic view of an exemplary cost model view receivinginputs for a material cost, a lower specification limit scrappedviolation cost and an upper specification limit scrapped violation cost.

FIG. 20B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 20C provides a detail view of FIG. 20B.

FIG. 21A is a schematic view of an exemplary cost model view receivinginputs for a material cost, a lower specification limit scrappedviolation cost and an upper specification limit rework violation cost.

FIG. 21B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 21C provides a detail view of FIG. 21B.

FIG. 22A is a schematic view of an exemplary cost model view receivinginputs for a material cost and a lower specification limit reworkviolation cost.

FIG. 22B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 22C provides a detail view of FIG. 22B.

FIG. 23A is a schematic view of an exemplary cost model view receivinginputs for a material cost, a lower specification limit rework violationcost and an upper specification limit scrapped violation cost.

FIG. 23B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 23C provides a detail view of FIG. 23B.

FIG. 24A is a schematic view of an exemplary cost model view receivinginputs for a material cost, a lower specification limit rework violationcost and an upper specification limit rework violation cost.

FIG. 24B provides an exemplary graphical view of costs, probabilitydensity function of a set of production data, and an optimal data meanprovided by a real-time SPC system.

FIG. 24C provides a detail view of FIG. 24B.

FIGS. 25A and 25B are schematic views of exemplary cost model viewsreceiving cost model inputs.

FIGS. 25C-25E are exemplary graphical views of costs, probabilitydensity functions of a set of production data, and an optimal data meansprovided by a real-time SPC system.

FIGS. 26A and 26B are schematic views of an exemplary cost model viewreceiving cost model inputs.

FIG. 26C is an exemplary graphical view of a cross-correlation of costmodel inputs with production data.

FIG. 26D provides a detail view of FIG. 26C.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

When considering an improvement for a step in a manufacturing process, astatistical process control (SPC) practitioner may consider, at least inpart, a statistical analysis that may include a current mean for eachset of variable data used to monitor the quality of the step's output.SPC may be broadly broken down into three sets of activities:understanding a process; understanding causes of variation in theprocess; and eliminating or modifying sources that cause the variation.

Referring to FIG. 1, in some implementations, a real-time statisticalprocess control (SPC) system 100 (also referred to herein as a costinspector) includes a function module 110 that receives a cost model 112associated with at least one process step, and a data module 120. Thefunction module 110 determines a cost function 114 (e.g., a per-unitcost function) and a percentage-of-acceptable-parts function 116associated with executing the least one process step. The data module120 receives production data 122 (also referred to as variable data)corresponding to a measured quality metric of the executed process stepand determines a probability density function 124 for the receivedproduction data 122. A cross-correlation module 130 cross-correlates thecost function 114 with the probability density function 124 of theproduction data 122 to provide a cross-correlation 134 (a firstcross-correlation 134 a). In some examples, the cross-correlation module130 cross-correlates the percentage-of-acceptable-parts function 116with the probability density function 124 of the production data 122 toprovide another cross-correlation 134 (a second cross-correlation 134b). In some implementations, a cost module 140, executing on a computingprocessor, determines an average effective per-unit cost 144 to producea resultant of the at least one process step by dividing the firstcross-correlation 134 a of the cost function 114 and the probabilitydensity function 124 of the production data 122 by the secondcross-correlation 134 b of the percentage-of-acceptable-parts function116 and the probability density function 124 of the production data 122.

Referring to FIGS. 2-4B, in some implementations, the SPC system 100applies the cost model 112 to each set of received variable data 122used to monitor the quality of a respective manufacturing output of aprocess step 210 of a process 200 (e.g., a manufacturing process) todetermine an optimal mean 150 (e.g., via the cost module 140) for eachset of variable data 122. The optimal mean 150 for a set of variabledata 122 can be an average that represents an output of acceptablequality at the lowest cost. Recommended changes to the process 200,based in part on the calculated optimal mean 150, can be designed toshift a current mean for a set of variable data 122 to its optimalposition. The cost model 113 may include user-supplied cost factors 113which are applied to the variable data 122 to determine cost-relatedanalytics that can be used to quantify how much a processed unit costsand whether that unit can be processed more cheaply. The user-suppliedcost factors 113 may include a process step cost 113 a, a material cost113 b, a lower specification limits (LSL) violation scrap cost 113 c, alower specification limits (LSL) violation rework cost 113 d, an upperspecification limits (USL) violation scrap cost 113 e, and/or an upperspecification limits (USL) violation rework cost 113 f.

The process step cost factor 113 a can be used to calculate costs whichdo not fluctuate during a process step 210. These costs may be indirector fully loaded costs and can include such costs as building costs,machine costs and personnel costs. In a filling process step 210, forexample, running the filling machine costs a set amount regardless ofwhether containers exit the process step under-filled,acceptably-filled, or over-filled. Unlike the other cost factors, thisfactor is independent of the number of sets of variable data 122associated with a process step 210. In other words, a single processstep cost can be used for each set of variable data 122 used to monitora particular process step 210. Regardless of how many types ofmeasurements are taken for a process step 210, the indirect costs forthat step remains unchanged. In some implementations, the process stepcost factor 113 a is a per part factor that reflects the indirect costsincurred by one part passing through the process step 210.

The material cost factor 113 b can be used to calculate the cost ofmaterial consumed during a process step 210. The material cost can be adirect cost. Each set of variable data 122 used to monitor a processstep 210 can have its own material cost factor 113 b. The material costfactor 113 b may be a per unit of measure factor, meaning the valuereflects the per unit of measure cost for the material consumed. Forexample, the material cost factor 113 b of filling a container with 250ml of material may be the cost for one milliliter of the material. Forprocess steps 210 monitored by multiple sets of variable data, only onematerial cost factor 113 b may be considered for a process step thatconsumes only one type of material or multiple material cost factors 113b for multiple types of corresponding material.

The lower specification limit violation scrap cost factor 113 c can beused to calculate scrap costs related to violations of a lowerspecification limit (LSL). Since each set of variable data 122 used tomonitor a process step 210 can have an LSL, each set of variable data122 can have a corresponding lower specification limit violation scrapcost factor 113 c. In some implementations, the lower specificationlimit violation scrap cost factor 113 c includes two components: 1) acost to scrap a part and 2) a percentage of parts which are allowed toviolate the LSL without being scrapped. In some examples, the SPC system100 determines the cost to scrap a part using the material cost factor113 b supplied for the respective set of variable data 122 and theprocess step cost factor 113 a supplied for the process step 210monitored by the set of variable 122.

The lower specification limit violation rework cost factor 113 d can beused to calculate rework costs related to violations of a lowerspecification limit (LSL). Since each set of variable data 122 used tomonitor a process step 210 can have its own lower specification limit,each set of variable data 122 can have its own lower specification limitviolation rework cost factor 113 d. Consequently, the lowerspecification limit violation rework cost factor 113 d can be a pervariable factor that reflects the cost to rework an aspect of a partmeasured by the set of variable data 122 for which an LSL violation wasrecorded.

The upper specification limit violation scrap cost factor 113 e can beused to calculate rework costs related to violations of an upperspecification limits (USL). Each set of variable data 122 can have acorresponding USL violation scrap cost factor 113 e. In someimplementations, the USL violation scrap cost factor 113 e includes twocomponents: 1) the cost to scrap a part and 2) a percentage of partswhich are allowed to violate the USL without being scrapped. In someexamples, the SPC system 100 determines the cost to scrap a part usingthe material cost factor 113 b supplied for the respective set ofvariable data 122 and the process step cost factor 113 a supplied forthe process step 210 monitored by the set of variable 122.

The upper specification limit violation rework cost factor 113 f can beused to calculate rework costs related to violations of an upperspecification limit (USL). Since each set of variable data 122 used tomonitor a process step 210 can have its own USL, each set of variabledata 122 can have its own USL violation rework cost factor 113 f.Consequently, the USL violation rework cost factor 113 f can be a pervariable factor that reflects the cost to rework an aspect of a partmeasured by the set of variable data 122 for which a USL violation wasrecorded.

The cost model 112 may include one or more process steps 210 with one ormore associated cost factors 113 for each process step 210. In theexample shown in FIG. 2, the cost model 112 includes a single processstep 210, such as a filling step, with associated cost factors 113measured by one set of variable data 122, such as by net weight for thefilling step. In the example shown in FIG. 3, the cost model 112includes cost factors 113 for a single process step 210 measured bymultiple sets of variable data 122. For example, a drilling process stepcan be measured by a hole location and a hole diameter. In this example,the cost model 112 includes one process step cost factor 113 a and oneset of the applicable optional cost factors 113 for each set of variabledata 122. FIG. 4 illustrates a cost model 112 for multiple process steps210 within a manufacturing process (e.g., a bottling process with abottle pre-forming process step followed by a bottle injection moldingprocess step followed by a bottle filling process step). The cost model112 for a multi-step process may include a set of cost factors 113 foreach process step 210, where each set of cost factors 113 includes oneprocess step cost factor 113 a and one or more other cost factors 113b-113 f for each set of variable data 122 associated with the respectiveprocess step 210.

While cost models 112 consisting of a single process step 210 havevalue, cost models 112 comprising multiple process steps 210 account forthe accumulative costs of a process 200. In isolation, the costs of alater process step 210 in a process may be thought to be less than thecosts of an earlier step 210. However, a true cost of a later processstep 210 may be the sum of the costs unique to that process step 210 andthe costs of each earlier process step 210 in the process 200. Forexample, the true cost of scrapping a part at the end of a process 200can be the time and materials of every process step 210 invested in thepart. In a multi-step cost model 112, the SPC system 100 may carryearlier costs forward and appropriately incorporate them into thecalculation of later optimal means.

In some implementations, the cost model 112 includes one process stepcost factor 113 a for each process step 210 and, for each set ofvariable data 122 associated with each respective process step 210, oneor more discretionary cost factors 113 (e.g., a material cost 113 b, alower specification limits (LSL) violation scrap cost 113 c, a lowerspecification limits (LSL) violation rework cost 113 d, an upperspecification limits (USL) violation scrap cost 113 e, and/or an upperspecification limits (USL) violation rework cost 113 f) relevant to thatset of variable data 122.

In some examples, there are ten cost factor arrangements of thediscretionary cost factors 113 b-113 f, where every set of variable data122 represented in the cost model 112 employs one of the cost factorarrangements. Table 1 illustrates the cost factor arrangements.

Exemplary machining process steps are suggested, but are not limited tothose listed.

TABLE 1 Cost Factor Arrangements LSL LSL USL USL Violation ViolationViolation Violation Material Scrap Rework Scrap Rework Exemplary ProcessSteps 1 X X Machining (with no opportunity for rework) 2 X X Machining(with opportunity for rework) 3 X X Drilling, boring 4 X X Forging, heattreating 5 X X Filling (with stated minimum) 6 X X X X Extrusion,injection molding, etching (with no opportunity for rework) 7 X XEtching (with opportunity for rework) 8 X X Joining (with opportunityfor rework) 9 X X X Plating or chemical deposition (with no opportunityfor rework) 10 X X X Plating or chemical deposition (with no opportunityfor rework)

FIGS. 4C and 4D illustrate an exemplary cost model user interface 400that allows a user to create a cost model 112 and define one or moreprocess steps 210 by assigning one or more cost factors 113 to eachprocess step 210. In the example shown, the process step 210 has lowerspecification limits (LSL) cost factors 410 and upper specificationlimits (USL) cost factors 420 available. The LSL cost factors 410 mayinclude process rules 412, such as whether to enforce a specificationlimit and/or whether violations are reworked. The LSL cost factors 410may include rework settings 414, such as a maximum rework attempts 414a, a rework threshold 414 b, a cost of each attempt 410 c, a chance ofsuccess 414 d, and/or a chance of scrap 414 e. The LSL cost factors 410may include final disposition settings 416, such as a percent violationsallowed 416 a, a disposal cost 416 b, a scrap recovery cost 416 c,and/or a material recovery cost 416 d. Similarly, the USL cost factors420 may include process rules 422, such as whether to enforce aspecification limit and/or whether violations are reworked. The USL costfactors 420 may include rework settings 424, such as a maximum reworkattempts 424 a, a rework threshold 424 b, a cost of each attempt 420 c,a chance of success 424 d, and/or a chance of scrap 424 e. The USL costfactors 420 may include final disposition settings 426, such as apercent violations allowed 426 a, a disposal cost 426 b, a scraprecovery cost 426 c, and/or a material recovery cost 426 d. Other costfactors in various arrangements are possible as well.

FIG. 5 provides an exemplary arrangement 500 of operations executable ona computer process for a method of real-time statistical processcontrol. The operations include determining 502 a per-unit cost function114 for executing a process step 210, determining 504 apercentage-of-acceptable-parts function 116 for executing a process step210, and receiving 506 production data 122 into memory. The productiondata 122 corresponds to a measured quality metric of the executedprocess step 210. The operations further include determining 508 aprobability density function 124 for the received production data 122,executing 510 on a processor a correlation routine (e.g., using thecross-correlation module 130) for cross-correlating the per-unit costfunction 114 with the probability density function 124 of the productiondata 122 to provide a first cross-correlation 134 a, executing 512 onthe processor the correlation routine for cross-correlating thepercentage-of-acceptable-parts function 116 with the probability densityfunction 124 of the production data 122 to provide a secondcross-correlation 134 b, and determining 514 an average effectiveper-unit cost 144 to produce a resultant of the process step 210 bydividing the first cross-correlation 134 a by the secondcross-correlation 134 b.

In some implementations, the operations include displaying on anelectronic device a plot corresponding to the determined averageeffective per-unit cost 144 and/or determining an optimal process meanassociated with the plotted minimum average effective per-unit cost 150associated with the data distribution 124. The operations may includepresenting a process offset based at least in part on a differencebetween the optimal process mean 150 and a current process meanassociated with the data distribution 124. The process offset can beassociated with a cost savings of the process step 210. The minimumaverage effective per-unit cost 150 can be determined iteratively usingdifferent standard deviations for the probability density function 124of the received data 122 that corresponds to a theoretical reduction inprocess variability. The iteratively determined minimum averageeffective per-unit costs 150 can be presented electronically in agraphical format.

Referring to FIG. 6, in the following lathing operation example, whereincoming blanks are machined to a relatively smaller diameter, the costof an incoming blank is $2.50 and the cost to pass a lathe over theblank one time is $1.00. The upper specification limit (USL) for adiameter of the lathed blank is 1.001 inches. The lower specificationlimit (LSL) for the diameter of the lathed blank is 0.999 inches. If,after one pass by the lathe, the blank has a diameter in excess of theUSL, the blank is reworked in a secondary, manual process at a cost of$4.00. One pass through this secondary process results in a blank with adiameter that falls between the USL and LSL. If, after one pass by thelathe, the blank is found to have a diameter less than the LSL, theblank is scrapped. In this example, after one pass by the lathe, 60% ofthe blanks violate neither the USL nor LSL, 30% violate the USL, and 10%violate the LSL. If there were no specification limits to contend with,every finished blank would cost $3.50 ($2.50 for the blank and $1.00 forthe lathing). However, the USL of 1.001 inches forces some of the lathedblanks to be reworked resulting in a total per reworked unit cost of$7.50 ($2.50 for the blank, $1.00 for lathing and $4.00 for rework).Just as the USL forces some blanks to be reworked, the LSL of 0.999inches forces some of the lathed blanks to be scrapped. Determining thecost to assign to scrapped (i.e. bad) parts, however, is not asstraightforward as determining rework costs.

Rather than determining the cost of a bad part directly, the SPC system100 may determine the cost of a good part by distributing the total costof all bad parts over the total cost of all good parts. In the example,out of a hundred lathed parts, 10 will be bad, 60 will be good after onepass by the lathe, and another 30 will be good after the manualsecondary process. Since the 10 bad parts have the same initialprocessing costs ($1.00) and material costs ($2.50) as a good part, eachcan be said to cost $3.50, resulting in a total of $35.00. Distributingthe cost of the 10 bad parts over the cost of the 90 good parts resultsin each good part costing $3.50 plus 1/90*$35.00, which equals $3.89.The distributed cost can be referred to as an effective cost of a goodpart EC_(good).

EC_(good) =C _(good) /P _(good)  (1)

EC_(good) is the effective cost of one good part (accounting for scraplosses), C_(good) is the actual cost of one good part (time andmaterials only), and P_(good) is the proportion of good parts made (inthe above example, 0.90). The proportion of good parts can be estimatedas the percentage of all measurements in a distribution below a specificvalue (such as a lower specification limit), as given by the cumulativedensity function of the distribution. The cumulative density function ofany distribution is defined as:

$\begin{matrix}{{{cdf}(z)} = {\int_{- \infty}^{Z}{p{{f(x)}}{x}}}} & (2)\end{matrix}$

where pdf(x) is the probability density function of interest.Integrating the probability density function results in:

P _(good)=1−cdf(LSL)  (3)

and

EC_(good) =C _(good) /P _(good) =C _(good)/1−cdf(LSL)  (4)

Equation 3 applies specifically to the lathing example, an operationwhere scrapping occurs at the lower specification limit and reworkoccurs at the upper specification limit.

The per unit rework cost can be determined by calculating a weightedaverage manufacturing cost. Instead of ignoring the rework costs, as wasdone for the preceding calculation, scrap costs will be temporarilyignored for this calculation. As a result, if 70% of the lathed blanksare at or below the USL, then these lathed blanks each have a per unitcost of $3.50, and the remaining 30% of the lathed blanks have a perunit cost of $7.50. The weighted average cost is:

C _(good)=0.70*$3.50+0.30*$7.50=$4.70  (5)

The general form of this calculation can be expressed as the integral ofthe value function (the per piece cost as illustrated in FIG. 5)multiplied by the weighting function (the probability density functionof the measurements) over all possible measurement values.

$\begin{matrix}{{\overset{\_}{C}}_{good} = {\int_{- \infty}^{\infty}{{{C_{good}(x)} \cdot p}{{f(x)}}{x}}}} & (6)\end{matrix}$

The formula for the average, effective cost of one good part becomes thefollowing, where the denominator is the specific form of P_(good)derived in this example:

$\begin{matrix}{{\overset{\_}{EC}}_{good} = {\frac{{\overset{\_}{C}}_{good}}{P_{good}} = \frac{\int_{- \infty}^{\infty}{{{C_{good}(x)} \cdot p}{{f(x)}}{x}}}{1 - {{cdf}({LSL})}}}} & (7)\end{matrix}$

For the lathing example, (i.e., 10% violating the LSL, 30% violating theUSL and 60% violation neither the LSL nor USL), the average, effectiveper piece cost is $5.22.

In addition to determining the average, effective per piece cost of apart produced by a particular process 200, the SPC system 100 can, insome implementations, project how a part can be processed more cheaply.The SPC system 100 calculates the average, effective per piece cost of apart based on the current mean of the corresponding variable data 122.For calculating the optimal mean, the SPC system 100 considers anadditional term, which introduces a shift of the process mean from itscurrent position to an optimal position. This is representedmathematically as:

$\begin{matrix}{{{\overset{\_}{EC}}_{good}(\delta)} = {\frac{{\overset{\_}{C}}_{good}}{P_{good}} = \frac{\int_{- \infty}^{\infty}{{{C_{good}(x)} \cdot p}{{f\left( {x + \delta} \right)}}{x}}}{1 - {{cdf}\left( {{LSL} - \delta} \right)}}}} & (8)\end{matrix}$

The numerator of the function is a cross-correlation of the cost tomanufacture a good part and the probability density function. Thefunction's minimum can be determined by solving the following equationfor δ:

$\begin{matrix}{{\frac{}{\delta}{{\overset{\_}{EC}}_{good}(\delta)}} = 0} & (9)\end{matrix}$

Once the minimum, δ_(opt), is found, the optimal mean, μ_(opt), (i.e.,the process mean having the lowest average, effective per piece cost)can be given by:

μ_(opt)=μ₀+δ_(opt)  (10)

where, μ₀ is the current process mean as determined from the variabledata 122 (i.e., production data) under analysis, δ_(opt) is the solutionto equation 9. (Note that the solution may be constrained such thatLSL≦μ₀+δ_(opt)≦USL.) With the optimal mean located, the minimum cost(the lowest average, effective per piece cost) can be given by:

C _(opt)= EC _(good)(δ_(opt))  (11)

The average, effective per piece cost can be evaluated at δ_(opt).

In some implementations, the minimum, average, effective per unitmanufacturing cost is given by:

$\begin{matrix}{{\frac{}{\delta}{{\overset{\_}{EC}}_{good}\left( {\mu_{0} + \delta} \right)}} = 0} & (12)\end{matrix}$

The value of which δ satisfies the expression may be the value at whichEC _(good)(μ+δ) reaches its minimum value. EC _(good)(·) is the averageper unit manufacturing cost at any process set point and can be definedas:

$\begin{matrix}{{{\overset{\_}{EC}}_{good}\left( {\mu_{0} + \delta} \right)} = \frac{R_{xy}(\delta)}{P_{good}(\delta)}} & (13)\end{matrix}$

where, μ₀ is the current process mean as determined from the variabledata 122 (i.e., production data), δ is any real number such thatLSL≦(μ₀+δ)≦USL (e.g., to constrain the recommended process set point tolay between the specification limits.)

R_(xy)(δ) is the cross-correlation between two real-valued functions andmay be defined as:

$\begin{matrix}{{R_{xy}(\delta)} = {\int_{- \infty}^{\infty}{{{C_{good}\left( {\zeta + \delta} \right)} \cdot p}{{f(\zeta)}}{\zeta}}}} & (14)\end{matrix}$

where, C_(good)(·) is the per unit manufacturing cost (i.e., not theaverage per unit manufacturing cost) at a given set point, pdf(·) is theprobability density function (e.g., one of twelve Pearson distributions)that best fits the variable data 122 analyzed.

P_(good)(δ) is the proportion of good parts made when running at aprocess set point of μ₀+δ and is given by:

P _(good)(δ)=1−ScrapTail_(L)(δ)−ScrapTail_(U)(δ)  (15)

where,

Allowance_(U) is the percentage of parts which violate the USL that areallowed to escape detection without consequence.

Allowance_(L) is the percentage of parts which violate the LSL that areallowed to escape detection without consequence.

$\begin{matrix}{{{ScrapTail}_{L}(\delta)} = \left\{ \begin{matrix}{{{cdf}\left( {{LSL} - \delta} \right)} - {{Allowance}_{L}({if\_ scrapping})}} \\{0\_ ({otherwise})}\end{matrix} \right.} & (16) \\{{{ScrapTail}_{U}(\delta)} = \left\{ \begin{matrix}{{{cdf}\left( {{USL} - \delta} \right)} - {{Allowance}_{U}({if\_ scrapping})}} \\{0\_ ({otherwise})}\end{matrix} \right.} & (17) \\{{{cdf}(z)} = {\int_{- \infty}^{Z}{{{pdf}(x)}{x}}}} & (18)\end{matrix}$

pdf(x) is the probability density function (e.g., one of twelve Pearsondistributions) that best fits the variable data 122 analyzed.

C _(good)(μ)=ProcessStepCost+LowerReworkCost(μ)+UpperReworkCost(μ)  (19)

where, ProcessStepCost is a user-specified, constant, manufacturing costassociated with the respective process step 210.

LowerReworkCost(μ)=C _(ReworkAttempt) _(L) ·μ(LSL−μ)  (20)

UpperReworkCost(μ)=C _(ReworkAttempt) _(U) ·μ(μ−USL)  (21)

C_(ReworkAttempt) _(L) is a user-specified, constant cost associatedwith reworking one unit that violates the lower specification limit.C_(ReworkAttempt) _(U) is a user-specified, constant cost associatedwith reworking one unit that violates the upper specification limit. LSLis a user-specified, lower specification limit of the variable data 122(e.g., production data) being analyzed. USL is a user-specified, upperspecification limit of the variable data 122 (e.g., production data)being analyzed. μ(·) is a standard unit step function.

In some implementations, the SPC system 100 executes a cost-benefitanalysis for reducing process variability. Qualitatively, manufacturingprocesses have some amount of piece-to-piece variation. An equipmentupgrade may engender a significant reduction in such variation.Equipment upgrades, however, come at a cost and this immediately raisesthe question of return. The SPC system 100 may provide predictiveanalyses to determine a return-on-investment for equipment upgrades(e.g., to reduce process variability). After determining the optimalmean and minimum average cost values for the current process 200, theoptimal mean and minimum average cost values can be recalculated using amodified distribution. The probability density function that wasoriginally fit to the production data can be adjusted to have a smallerstandard deviation. In some examples, the SPC system 100 uses 90% of theoriginal standard deviation, then 80%, then 70%, and so on, all the waydown to a standard deviation of 0. (With a standard deviation of zero,the distribution collapses into a Dirac delta function—a uni-valuedprobability density function with unity area.)

While it is theoretically possible to express C_(opt) as a function ofstandard deviation, the resulting function may only be evaluated for afew of the possible probability density functions. As a result, C_(opt)may be evaluated numerically at discrete values of σ and then plottedfor analysis.

FIG. 7A graphically illustrates a determined cost per unit versusprocess set point of a process 200. In the example shown, the currentprocess mean, μ₀, is 9.000 and the optimal process mean, μ_(opt), hasbeen calculated to be 8.979. The graph provides a histogram of thevariable data 122 analyzed and provides plots of C_(good), EC_(good)(δ), and pdf(x) (which is the fitted probability densityfunction). FIG. 7B provides a detail view of FIG. 7A. The bottomhorizontal line in this detail view represents the $3.50 unit cost. Thecost of process offset point near the top of the chart represents thiscost plus the cost of variability at 100% of the current process sigmaσ. Reducing the exemplary process 200 by 10% results in cutting the lossdue to variability almost in half and reducing the process sigma σ by30% results in the near elimination of all loss due to processvariability.

The following process variability sensitivity (PVS) ratio evaluates thepercent reduction in average, effective per piece cost that can beexpected by decreasing the process sigma σ by 10%:

$\begin{matrix}{{PVS} = \frac{C_{opt} - C_{{opt} - {90\%}}}{C_{opt}}} & (22)\end{matrix}$

In the lathing example, PVS=2.033. By reducing the process sigma σ by10%, the average, effective per piece cost decreases by 2.033% (from$3.6743/unit to $3.5996/unit), resulting in a savings of 7.47cents/unit.

While the graph of a Taguchi Loss curve may look somewhat like a SPCsystem curve, the calculations involved are fundamentally different. TheSPC system 100 makes no assumptions about the cost of scrap, nor thedistribution of the measurements being analyzed. The Taguchi Lossmethod, on the other hand, begins with the assumption that the lowestper piece cost occurs at the specified target or nominal value. It thenassumes that the cost function is parabolic; that is, it assumes the perpiece manufacturing cost increases quadratically as the distance fromtarget increases. Finally, because of the assumed parabolic shape of theper piece manufacturing cost curve, the Taguchi Loss method tacitlyassumes the cost function is symmetric on either side of the specifiedtarget. While all of these assumptions can be defended in certainapplications, they introduce significant short comings when are appliedto real-world production data for the purpose of calculating actual,realized costs. The Taguchi Loss method is often used to model futuremanufacturing costs at the design stage of product development, but isnot the best model to calculate actual costs accrued during production.

A graphical representation of a per piece cost is not necessarilyparabolic, as assumed in the Taguchi Loss method, which makes anapproximation of a loss function that would (at some future date) bemeasurable once production started. Since there is no way to accuratelypredict exactly what the equation of that loss function would ultimatelybe, the Taguchi Loss method uses the first two terms from a Taylorseries expansion of a generic function as an estimate of the actual costfunction. This may be a good model to use when production data is notavailable. However, a Taylor series is itself only an approximation ofanother function and is only somewhat accurate near the point aroundwhich the expansion is made. One of the strengths of the SPC system 100is that is can be used to calculate the actual per piece manufacturingcost based on user-specified scrapping and rework rules and a set ofempirically collected variable data 122 (e.g., production data), asrepresented by equation 13, for example.

In some examples, the SPC system 100 may indicate that the lowest perpiece manufacturing cost point may not be at an obvious target. Forexample, a process 200 that requires scrapping a below-spec unit, butallows reworking an above-spec unit, in many cases it is probable thatthe lowest cost point will be above the specified target value,simultaneously increasing the probability of rework and decreasing theprobability of scrap.

Referring to FIGS. 8-12, in some implementations, a user of the SPCsystem 100 can set up a cost model 112 by opening an administratorwindow 800, right-click on a part 810 for which to set up a cost model112 to open an options menu 820 and select properties 830, as shown inFIG. 8. (It does not matter at this point whether the cost model is tobe a single-process step cost model or a multi-process step cost model.)The SPC system 100 displays a cost model part properties dialog box 900having a process pane 910 that can be populated with the process steps210 of a particular process 200. The user may add and remove processsteps 210 to the process pane 910 by clicking or selecting a createprocess step button 920 a and remove process step button 920 b,respectively. For each selected process step 210, the user may overwritea default name provided in a process step name box 930 with adescriptive name, such as “filling,” for the process step 210, as shownin FIG. 9B. Moreover, the user may specify a process step cost factor113 a for the selected process step in a process step cost factor box940. Next, the user may assign or remove sets of variable data 122 tothe selected process step 210 by selecting one or more sets of variabledata 122 in an available variables pane 950 and selecting an addvariables button 960 a or a remove variables button 960 b, respectively,as shown in FIG. 9C. As a result, the selected set(s) of variable data122 are assigned to the corresponding process step 210, as shown in FIG.9D. The user can add all available sets of variable data 122 in anavailable variables pane 950 by selecting an add all variables button960 d. Moreover, the user can remove all sets of variable data 122associated with the selected process step 210 by selecting a remove allvariables button 960 d. The user can reorder the process steps byselecting the move process step up button 970 a or the move process stepdown button 970 b.

The user may supply any cost factors 113 relevant to each set ofvariable data 122 used to monitor each respective process step 210.Referring to FIG. 10, if the set of variable data 122 has a materialcost factor 113 b, the user may enter a per unit of measure materialcost in a material cost box 1010 of a material cost pane 1000.

Referring to FIG. 11, which illustrates an exemplary lower specificationlimit violation pane 1100, for each set of variable data 122, the usermay select whether lower specification limit violations are ignoredusing an enforce/ignore radio button 1110, whether the part is scrappedby selecting a scrap radio button 1120 and assigning an associated lowerspecification limit violation scrap cost factor 113 c in a LSL scrapcost factor box 1130 (e.g., percent violations allowed), and/or whetherthe part is reworked by selecting a rework radio button 1140 andassigning an associated lower specification limit violation rework costfactor 113 d in a LSL rework cost factor box 1150. For the scrappedoption, the value entered is the percentage of violations allowed,which, in other words, is the percentage of parts which can violate theLSL without having to be scrapped. For the reworked option, the valueentered is the cost to bring one LSL-violating part into conformance.

Referring to FIG. 12, which illustrates an exemplary upper specificationlimit violation pane 1200, for each set of variable data 122, the usermay select whether upper specification limit violations are ignoredusing an ignore radio button 1210, whether the part is scrapped byselecting a scrap radio button 1220 and assigning an associated upperspecification limit violation scrap cost factor 113 e in a USL scrapcost factor box 1230, and/or whether the part is reworked by selecting arework radio button 1240 and assigning an associated upper specificationlimit violation rework cost factor 113 f in a USL rework cost factor box1250. For the scrapped option, the value entered is the percentage ofviolations allowed, which, in other words, is the percentage of partswhich can violate the USL without having to be scrapped. For thereworked option, the value entered is the cost to bring oneUSL-violating part into conformance.

Once a cost model 112 is set up in the SPC system 100, the cost model112 can be applied to each set of variable data 122 represented. Byapplying a cost model 112 to a constituent set of variable data 122, anoptimal mean for that set of variable data 122 can be calculated andviewed. In the example shown in FIG. 13, a selected cost model 112 canbe applied to a constituent set of variable data 122 by displaying thatset of variable data 122 in the SPC system 100. For example, in theadministrator window 800, the user may right-click on the set ofvariable data 122 to open the options menu 820 and select query 840 toexecute a variable analyzer 1400, an example of which is shown in FIG.14, on the selected set of variable data 122. The variable analyzer 1400can display a histogram of the set of variable data 122 as well as acorresponding optimal mean.

In some implementations, the SPC system 100 provides the current processmean as the average of the set of variable data 122 as of the lastmeasurement recorded (e.g., last real-time measurement of the process200) and the average cost per unit, which reflects the average totalcost to manufacture one unit at the current mean. The average cost perunit may be a composite of process step costs, material costs, scrapcosts and rework costs. The SPC system 100 may provide a minimum averagecost per unit and/or an optimal process mean, which can be a projectionof the average total cost that would be incurred at the optimal mean orthe lowest cost that can be expected given the present cost factors. TheSPC system 100 may provide a process offset as the difference betweenthe current process mean and optimal process mean for a given set(s) ofvariable data 122. Moreover, the SPC system 100 may provide a cost ofprocess offset as the difference between the current average cost perunit and the minimum average cost per unit.

In some implementations, the SPC system 100 provides information on howto capitalize on the cost-cutting opportunity associated with theoptimal mean for a set of variable data 122, by offering improvements tothe corresponding process step 210 to shift the process mean for the setof variable data 122 from its current position toward its optimalposition. Effecting such an improvement may require an intimateknowledge of the process step 210 and, in some cases, a familiarity withthe overall manufacturing process 200 of which the process step 210 is amember. For example, if the process step 210 is a filling process stepexecuted on a filling machine and the optimal process mean is less than(e.g., to the left of) the current process mean, the filling machine isover-filling and needs to be adjusted to fill containers with lessmaterial. If the optimal process mean is greater than (e.g., to theright of) the current process mean, the filling machine is under-fillingand needs to be adjusted to fill containers with more material. Inanother example, if the process step 210 includes dimension-reducing, aswith machining operations such as milling, and the optimal process meanis to the left of the current process mean, the process step 210 isunder-reducing and needs to be adjusted to further reduce the dimension.If the optimal process mean is to the right of the current process mean,the process step 210 is over-reducing and needs to be adjusted to reducethe dimension less. These examples show that the position of the optimalprocess mean relative to the position of the current process mean can beused to determine if a process step 210 is having too little effect(e.g., heating too little, plating too little, etc.) or too much effect(e.g., cooking too much, etching too much, etc.). Consequently, theprocess step 210 can be adjusted to correct the deficiency or excess.For a set of variable data where the optimal process mean equals thecurrent process mean, the operation can be considered as optimized as itcan be with the current equipment. Further cost-cutting in this scenariomay only possible by upgrading the machine.

Referring to FIG. 15A, in the following example, a cost model 112 is fora heat treating process step 210 is shown in the process pane 910 of thecost model part properties dialog box 900. The set of variable data 122associated with the heat treating process step 210 is hardness. Inaddition to the process step cost factor 113 a entered in the processstep cost factor box 940, the cost model 112 has a lower specificationlimit violation scrap cost factor 113 c entered in the SL scrap costfactor box 1150 and an upper specification limit violation scrap costfactor 113 e entered in the USL scrap cost factor box 1230.

As illustrated in FIG. 15B, applying the cost model 112 to an exampleset of hardness data reveals that the data's optimal process mean is tothe right of the current process mean and has a cost of process offsetof just under five cents per unit. For every million units heat treated,this adds up to a waste of $49,200.00. To cut this cost, since the unitof measurement for this heat treating step is indentation depth, theprocess step 210 needs to be adjusted to harden the units less, which inturn would result in the indentation depth shifting closer to the USLwith respect to current position.

In a detail view shown in FIG. 15C, the lowest cost point for the heattreating process step 210 is shown at a midpoint between the LSL andUSL. This is due to the fact that both the cost model and thedistribution of data are symmetrical.

FIGS. 16A-16C illustrate an example of a cost model 112 for a grindingprocess step 210 having a set of associated variable data 122 for athickness of a part. In addition to the process step cost factor 113 aentered in the process step cost factor box 940, the cost model 112 hasa lower specification limit violation scrap cost factor 113 c entered inthe LSL scrap cost factor box 1150 and an upper specification limitviolation rework cost factor 113 e entered in the USL rework cost factorbox 1250. Applying the cost model 112 to an example set of Thicknessdata, as shown in FIG. 16B, reveals that the data's optimal process meanis to the left of the current process mean and has a cost of processoffset of just under five cents per unit. For every million unitsground, this adds up to a waste of $47,600.00. In this example, thegrinding process step 210 is under-grinding. To cut costs, the grindingprocess step 210 needs to be adjusted to grind more. The detail viewshown in FIG. 16C illustrates that the optimal process mean will onlyapproach a target that is centered between the LSL and USL if processvariability is reduced.

FIGS. 17A-17C illustrate an example of a cost model 112 for a boringprocess step 210 having a set of associated variable data 122 for adiameter of a part. In addition to the process step cost factor 113 aentered in the process step cost factor box 940, the cost model 112 hasa lower specification limit violation rework cost factor 113 d enteredin the LSL rework cost factor box 1150 and an upper specification limitviolation scrap cost factor 113 e entered in the USL scrap cost factorbox 1230. Applying the cost model 112 to an example set of diameterdata, as shown in FIG. 17B, reveals that the data's optimal process meanis to the left of the current process mean and has a cost of processoffset of around one and a half cents per hole. For every million holesbored, this adds up to a waste of $16,000.00. The boring process step210 is boring excessively wide holes. To cut costs, the boring processstep 210 needs to be adjusted to bore holes of a lesser diameter. Thedetail view shown in FIG. 17C illustrates that no gains beside thoseassociated with process offset are possible. There are no savings, inother words, that can be expected from reducing variability becausethere is insufficient loss due to variability.

FIGS. 18A-18C illustrate an example of a cost model 112 for a forgingprocess step 210 having a set of associated variable data 122 for athickness of a part. In addition to the process step cost factor 113 aentered in the process step cost factor box 940, the cost model 112 hasa lower specification limit violation rework cost factor 113 d enteredin the LSL rework cost factor box 1150 and an upper specification limitviolation rework cost factor 113 f entered in the USL rework cost factorbox 1250. Applying the cost model 112 to an example set of thicknessdata, as shown in FIG. 18B, reveals that the data's optimal process meanis to the left of the current process mean and has a cost of processoffset of a little over four cents per unit. For every million unitsforged, this adds up to a waste of $43,900.00. The forging process step210 is forging units with too great a thickness. To cut costs, theprocess step needs to be adjusted so the forged units are thinner. Thedetail view shown in FIG. 18C illustrates that the cost of processoffset is over ten times the cost of process variability. Consequently,the focus for improvement should be shifting the current mean to theoptimal mean. Once that is done, further gain from reducing processvariability can be pursued if, in light of other priorities, it makessense.

FIGS. 19A-19C illustrate an example of a cost model 112 for a fillingprocess step 210 in which a machine dispenses material into containers.The set of variable data 122 associated with the filling process step210 is net weight. In addition to the process step cost factor 113 aentered in the process step cost factor box 940, the cost model 112 hasa material cost factor 113 b entered in the material cost factor box1010 and a lower specification limit violation scrap cost factor 113 centered in the LSL scrap cost factor box 1150. Applying the cost model112 to an example set of net weight data, as shown in FIG. 19B, revealsthat the data's optimal process mean is to the right of the currentprocess mean and has a cost of process offset of just over two cents percontainer. For every million containers filled, this adds up to a wasteof $22,200.00. The filling process 210 step is over-filling and needs tobe adjusted to fill containers with less material. The detail view shownin FIG. 19C illustrates that the cost of process variability is nearlyfive times the cost of process offset. As a result, initial improvementscan be directed at process variability. In addition, the processvariability sensitivity shows that the cost of a finished unit can bereduced by 0.02% for every 10% decrease in the process variability. Theevenly spaced dots between the LSL and optimal process mean illustratethis linear progression of cost savings.

FIGS. 20A-20C illustrate an example of a cost model 112 for an injectionmolding process step 210 having a set of associated variable data 122for a diameter of a part. In addition to the process step cost factor113 a entered in the process step cost factor box 940, the cost model112 has a material cost factor 113 b entered in the material cost factorbox 1010, a lower specification limit violation scrap cost factor 113 centered in the LSL scrap cost factor box 1150 and an upper specificationlimit violation scrap cost factor entered in the USL scrap cost factorbox 1230. Applying the cost model 112 to an example set of diameterdata, as shown in FIG. 20B, reveals that the data's optimal process meanis to the left of the current process mean and has a cost of processoffset of over thirty-two dollars per unit. For every million unitsmolded, this adds up to a waste of $32,489,400.00. The injection moldingprocess step 210 is producing molds with too great a diameter and needsto be adjusted to produce molds of relatively lesser diameter to cutcosts. The detail view shown in FIG. 20C illustrates that a “build forLSL scrap” process setting makes sense since, in the end, this willresult in lowest average effective cost per good unit.

FIGS. 21A-21C illustrate an example of a cost model 112 for a castingprocess step 210 having a set of associated variable data 122 for adiameter of a part. In addition to the process step cost factor 113 aentered in the process step cost factor box 940, the cost model 112 hasa material cost factor 113 b entered in the material cost factor box1010, a lower specification limit violation scrap cost factor 113 centered in the LSL scrap cost factor box 1150 and an upper specificationlimit violation rework cost factor 113 f entered in the USL rework costfactor box 1250. Applying the cost model 112 to an example set ofdiameter data, as shown in FIG. 21B, reveals that the data's optimalprocess mean is to the right of the current process mean and has a costof process offset of just under two and a half dollars per unit. Forevery million units cast, this adds up to a waste of $2,496,400.00. Thecasting process step 210 is producing casts with diameters that are toosmall and needs to be adjusted to produce casts having relatively largerdiameters to cut cost. The detail view shown in FIG. 21C illustratesthat putting a little more material into each mold can have thedesirable but counter-intuitive effect of decreasing the cost/unit.

FIGS. 22A-22C illustrate an example of a cost model 112 for a fillingprocess step 210 in which a machine dispenses material into a container.A set of variable data 122 associated with the process step 122 is for anet weight. In addition to the process step cost factor 113 a entered inthe process step cost factor box 940, the cost model 112 has a materialcost factor 113 b entered in the material cost factor box 1010 and alower specification limit violation rework cost factor 113 d entered inthe LSL rework cost factor box 1150. Applying the cost model 112 to anexample set of net weight data, as shown in FIG. 22B, reveals that thedata's optimal process mean is just slightly to the left of the currentprocess mean and has a cost of process offset of less than a half-centper unit. For every million containers filled, this adds up to a wasteof $3,600.00. The filling process step 210 is slightly overfillingcontainers and can be adjusted to fill the containers with less materialto save costs. The detail view shown in FIG. 22C illustrates thatnegligible gain will be had by shifting the current process mean butthat considerable gain can be had by decreasing process variability.

FIGS. 23A-23C illustrate an example of a cost model 112 for a chemicaldeposition process step 210 having a set of associated variable data fora thickness. In addition to the process step cost factor 113 a enteredin the process step cost factor box 940, the cost model 112 has amaterial cost factor 113 b entered in the material cost factor box 1010,a lower specification limit violation rework cost factor 113 d enteredin the LSL rework cost factor box 1150 and an upper specification limitviolation scrap cost factor 113 e entered in the USL scrap cost factorbox 1230. Applying the cost model 112 to an example set of thicknessdata, as shown in FIG. 23B, reveals that the data's optimal process meanis to the left of the current process mean and has a cost of processoffset of just over a hundred and eighty-three dollars per unit. Forevery million units processed, this adds up to a waste of$183,378,300.00. The chemical deposition process step 210 isover-depositing chemicals and needs to be adjusted to deposit relativelyless material to save costs. The detail view shown in FIG. 23Cillustrates that the cost of process offset is 30 times the cost ofprocess variability, which dictates that a priority improvement isshifting the process mean.

FIGS. 24A-24C illustrate an example of a cost model 112 for a platingprocess step 210 having a set of associated variable data 122 forthickness. In addition to the process step cost factor 113 a entered inthe process step cost factor box 940, the cost model 112 has a materialcost factor 113 b entered in the material cost factor box 1010 and alower specification limit violation rework cost factor 113 d entered inthe LSL rework cost factor box 1150 and an upper specification limitviolation rework cost factor 113 f entered in the USL rework cost factorbox 1250. Applying the cost model 112 to an example set of thicknessdata, as shown in FIG. 23C, reveals that the data's optimal process meanis to the right of the current process mean and has a cost of processoffset of eighty cents per unit. For every million units plated, thisadds up to a waste of $80,000.00. The plating process step 210 isunder-plating and needs to be adjusted to plate more heavily to savecosts. The detail view shown in FIG. 24C illustrates that the bulk ofloss comes from LSL rework and that loss increases much more slowly asthe USL is approached.

FIGS. 25A-25D illustrate an exemplary SPC system 100 displaying a costmodel 900 having a process pane 910 that can be populated with theprocess steps 210 of a particular process 200. The user may add andremove process steps 210 to the process pane 910 by clicking orselecting a create process step button 920 a and remove process stepbutton 920 b, respectively. The user may assign or remove sets ofvariable data 122 to the selected process step 210 by selecting one ormore sets of variable data 122 in an available variables pane 950 andselecting an add variables button 960 a or a remove variables button 960b, respectively. The user can add all available sets of variable data122 in an available variables pane 950 by selecting an add all variablesbutton 960 d. Moreover, the user can remove all sets of variable data122 associated with the selected process step 210 by selecting a removeall variables button 960 d. The user can reorder the process steps byselecting the move process step up button 970 a or the move process stepdown button 970 b. The user may enter a per unit of measure materialcost in a material cost box 1010 of a material cost pane 1000.

The LSL violation pane 1100 (FIG. 25A) allows the user to assign lowerspecification limit properties for each set of variable data 122. In theexample shown, the user can select an enforce/ignore radio button 1110to assign whether the lower specification limits are enforced orignored. A rework radio button 1140 allows the user to assign whether apart is reworked. A rework settings pane 1152 allows the user to assignassociated lower specification limit violation rework cost factors 113d. The rework settings pane 1152 includes a maximum rework attemptsradio button 1153 for assigning whether to enforce a maximum number ofrework attempts 1154. A rework threshold radio button 1155 allows theuser to assign whether to enforce a rework threshold 1156 a, a cost ofeach attempt 1156 b, a chance of success 1156 c, and a chance of scrap1156 d. A final disposition pane 1158 allows the user to assignassociated lower specification limit violation scrap cost factors 113 cand includes a percent violations allowed 1158 a, a disposal cost 1158b, a scrap recovery cost 1158 c, and a percent material recovery 1158 d.

The USL violation pane 1200 (FIG. 25B) allows the user to assign upperspecification limit properties for each set of variable data 122. In theexample shown, the user can select an enforce/ignore radio button 1210to assign whether the upper specification limits are enforced orignored. A rework radio button 1240 allows the user to assign whether apart is reworked. A rework settings pane 1252 allows the user to assignassociated upper specification limit violation rework cost factors 113f. The rework settings pane 1252 includes a maximum rework attemptsradio button 1253 for assigning whether to enforce a maximum number ofrework attempts 1254. A rework threshold radio button 1255 allows theuser to assign whether to enforce a rework threshold 1256 a, a cost ofeach attempt 1256 b, a chance of success 1256 c, and a chance of scrap1256 d. A final disposition pane 1258 allows the user to assignassociated upper specification limit violation scrap cost factors 113 eand includes a percent violations allowed 1258 a, a disposal cost 1258b, a scrap recovery cost 1258 c, and a percent material recovery 1258 d.

FIGS. 25C-25E illustrate exemplary cost inspector views 2500 providing agraphical representation 2502 of applying the cost model 112 of theexample shown to a set of variable data 122. The graphicalrepresentation 2502 is plotted as cost per unit versus process setpoints and includes input costs 2510, processing costs 2512, averagecost per unit 2514, cost versus mean 2516, current distribution 2518,and optimal distribution 2520.

The cost model parameters that can be considered in determining theminimum, average, effective per unit manufacturing cost include:

USL is the user-specified, upper specification limit of the productiondata being analyzed.

LSL is the user-specified, lower specification limit of the productiondata being analyzed.

ProcessStepCost is the total, loaded processing cost associated with themanufacture of one production unit.

PerUnitCost is the per unit of measure cost of material consumed duringthe production of a single unit.

N_(U) (0≦N_(U)<∞) is the maximum number of times that a unit violatingits upper specification limit will be reworked in an effort to correctthe violation.

N_(L) (0≦N_(L)<∞) is the maximum number of times that a unit violatingits lower specification limit will be reworked in an effort to correctthe violation.

URT (USL≦URT<∞) is the dimensional limit beyond which a rework attemptwill not be made. That is, a manufactured unit whose dimensional valueexceeds this amount will not be reworked.

LRT (−∞<LRT≦LSL) is the dimensional limit beyond which a rework attemptwill not be made. That is, a manufactured unit whose dimensional valueis less than this amount will not be reworked.

C_(ReworkAttempt) _(U) is the cost of a single rework attempt that seeksto correct a violation of a unit's USL.

C_(ReworkAttempt) _(L) is the cost of a single rework attempt that seeksto correct a violation of a unit's LSL.

p_(Success) _(U) is the percentage chance that any single rework attemptwill move the measured dimension of the unit between the specificationlimits. This parameter is the chance of correcting a violation of theUSL.

p_(Success) _(L) is the percentage chance that any single rework attemptwill move the measured dimension of the unit between the specificationlimits. This parameter is the chance of correcting a violation of theLSL.

p_(Scrap) _(U) is the percentage chance that any single rework attemptwill destroy the unit beyond repair. This parameter is the chance ofdestroying a unit currently in violation of its USL.

p_(Scrap) _(L) is the percentage chance that any single rework attemptwill destroy the unit beyond repair. This parameter is the chance ofdestroying a unit currently in violation of its LSL.

Allowance_(U) is the percentage of parts which violate the USL that areallowed to escape detection without consequence.

Allowance_(L) is the percentage of parts which violate the LSL that areallowed to escape detection without consequence.

DisposalCost_(U) specifies an addition per unit cost that must be paidto dispose of a unit that will be scrapped because of a USL violation.

DisposalCost_(L) specifies an addition per unit cost that must be paidto dispose of a unit that will be scrapped because of an LSL violation.

ScrapRecovery_(U) is the expected revenue received for one unit sold asscrap. This parameter is the per unit recovery value for those units inviolation of their USL.

ScrapRecovery_(L) is the expected revenue received for one unit sold asscrap. This parameter is the per unit recovery value for those units inviolation of their LSL.

MaterialRecovery_(U) specifies the percentage of material consumedduring manufacture that can be recovered before a unit is ultimatelyscrapped. This parameter applies to those units scrapped owing to aviolation of their USL.

MaterialRecovery_(L) specifies the percentage of material consumedduring manufacture that can be recovered before a unit is ultimatelyscrapped. This parameter applies to those units scrapped owing to aviolation of their LSL.

In some implementations, the minimum, average, effective per unitmanufacturing cost is given by the following:

$\begin{matrix}{{\frac{}{\delta}{{AEC}\left( {\mu_{0} + \delta} \right)}} = 0} & (23)\end{matrix}$

This is the standard way to find the x value(s) at which a functionreaches either a maximum or minimum. Because of our apriori knowledge ofthe shape of AEC(·), there can be more than one value of δ whichsatisfies this expression. Each value is found, and then the value of δwhich produces the minimum value of AEC(μ₀+δ) is selected.

AEC(·) is the average, effective, per unit manufacturing cost at anyprocess set point where the denominator is the general form of P_(good).It is defined as follows:

$\begin{matrix}{{{AEC}\left( {\mu_{0} + \delta} \right)} = {\frac{R_{xy}(\delta)}{R_{xz}(\delta)} = \frac{\int_{- \infty}^{\infty}{{{C_{good}\left( {\zeta + \delta} \right)} \cdot p}{{f(\zeta)}}{\zeta}}}{\int_{- \infty}^{\infty}{{{M_{good}\left( {\zeta + \delta} \right)} \cdot p}{{f(\zeta)}}{\zeta}}}}} & (24)\end{matrix}$

where,

μ₀ is the mean of the production data being analyzed;

δ is any real number such that LSL<μ₀+δ<USL (this constraint forces therecommended process set point to lay between the specification limits);

C_(good)(·) is the per unit manufacturing cost (not the average per unitmanufacturing cost) at a given set point and is defined below;

M_(good)(·) is the percentage of units manufactured at a given set pointthat are considered “good” or “saleable” (not the average percentage ofgood units manufactured) and is defined below; and

pdf(·) is the probability density function (e.g. one of the 12 Pearsondistributions) that best fits the data being analyzed.

Now, the per unit manufacturing cost is modeled by the followingpiecewise linear function,

$\begin{matrix}{{C_{good}(\mu)} = \left\{ {{\begin{matrix}{{m_{1} \cdot \mu} + b_{1}} & \left( {\mu < {LRT}} \right) \\{{m_{2} \cdot \mu} + b_{2}} & \left( {{LRT} \leq \mu < {LSL}} \right) \\{{m_{3} \cdot \mu} + b_{3}} & \left( {{LSL} \leq \mu < {USL}} \right) \\{{m_{4} \cdot \mu} + b_{4}} & \left( {{USL} < \mu \leq {URT}} \right) \\{{m_{5} \cdot \mu} + b_{5}} & \left( {{URT} < \mu} \right)\end{matrix}{where}},} \right.} & (25) \\{m_{1} = {{{Allowance}_{L} \cdot {PerUnitCost}} + {\left( {1 - {Allowance}_{L}} \right) \cdot \left( {1 - {MaterialRecovery}_{L}} \right) \cdot {PerUnitCost}}}} & (26) \\{b_{1} = {{{Allowance}_{L} \cdot {ProcessStepCost}} + {\left( {1 - {Allowance}_{L}} \right) \cdot \left( {{DisposalCost}_{L} - {ScrapRecovery}_{L} + {ProcessStepCost}} \right)}}} & (27) \\{m_{2} = {{p_{{ReworkSuccess}_{\; L}} \cdot m_{3}} + {\left( {1 - p_{{ReworkSuccess}_{\; L}}} \right) \cdot m_{1}}}} & (28) \\{b_{2} = {{p_{{ReworkSuccess}_{\; L}} \cdot \left( {b_{3} + C_{{ReworkSuccess}_{\; L}}} \right)} + {\left( {1 - p_{{ReworkSuccess}_{\; L}}} \right) \cdot \left( {b_{1} + C_{{{ReworkFailure}\;}_{L}}} \right)}}} & (29) \\{m_{3} = {PerUnitCost}} & (30) \\{b_{3} = {ProcessStepCost}} & (31) \\{m_{4} = {{p_{{ReworkSuccess}_{\; U}} \cdot m_{3}} + {\left( {1 - {p_{ReworkSuccess}}_{\; U}} \right) \cdot m_{5}}}} & (32) \\{b_{4} = {{p_{{ReworkSuccess}_{U}} \cdot \left( {b_{3} + C_{{ReworkSuccess}_{U}}} \right)} + {\left( {1 - p_{{ReworkSuccess}_{U}}} \right) \cdot \left( {b_{5} + C_{{ReworkFailure}_{U}}} \right)}}} & (33) \\{m_{5} = {{{Allowance}_{U} \cdot {PerUnitCost}} + {\left( {1 - {Allowance}_{U}} \right) \cdot \left( {1 - {MaterialRecovery}_{U}} \right) \cdot {PerUnitCost}}}} & (34) \\{{b_{5} = {{{Allowance}_{U} \cdot {ProcessStepCost}} + {\left( {1 - {Allowance}_{U}} \right) \cdot \left( {{DisposalCost}_{U} - {ScrapRecovery}_{U} + {ProcessStepCost}} \right)}}}{{And},}} & (35) \\{p_{{ReworkSuccess}_{L}} = {p_{{Success}_{L}} \cdot {\sum\limits_{k = 1}^{N_{L}}\; p_{{Rework}_{L}^{k - 1}}}}} & (36) \\{C_{{ReworkSuccess}_{L}} = {p_{{Success}_{L}} \cdot C_{{ReworkAttempt}_{L}} \cdot {\sum\limits_{k = 1}^{N_{L}}\; {k \cdot p_{{Rework}_{L}^{k - 1}}}}}} & (37) \\{C_{{ReworkFailure}_{L}} = {p_{{Scrap}_{L}} \cdot C_{{ReworkAttempt}_{L}} \cdot {\sum\limits_{k = 1}^{N_{L}}\; {k \cdot p_{{Rework}_{L}^{k - 1}}}}}} & (38) \\{{p_{{Rework}_{L}} = {1 - p_{{Success}_{L}} - p_{{Scrap}_{L}}}}{{Similarly},}} & (39) \\{p_{{ReworkSuccess}_{U}} = {p_{{Success}_{U}} \cdot {\sum\limits_{k = 1}^{N_{U}}\; p_{{Rework}_{U}^{k - 1}}}}} & (40) \\{C_{{ReworkSuccess}_{U}} = {p_{{Success}_{U}} \cdot C_{{ReworkAttempt}_{U}} \cdot {\sum\limits_{k = 1}^{N_{U}}\; {k \cdot p_{{Rework}_{U}^{k - 1}}}}}} & (41) \\{C_{{ReworkFailure}_{U}} = {p_{{Scrap}_{U}} \cdot C_{{ReworkAttempt}_{U}} \cdot {\sum\limits_{k = 1}^{N_{U}}\; {k \cdot p_{{Rework}_{U}^{k - 1}}}}}} & (42) \\{p_{{Rework}_{U}} = {1 - p_{{Success}_{U}} - p_{{Scrap}_{U}}}} & (43)\end{matrix}$

Finally, the percent good parts function is given by the followingpiecewise constant function,

$\begin{matrix}{{M_{good}(\mu)} = \left\{ {{\begin{matrix}p_{1} & \left( {\mu < {LRT}} \right) \\p_{2} & \left( {{LRT} \leq \mu < {LSL}} \right) \\p_{3} & \left( {{LSL} \leq \mu \leq {USL}} \right) \\p_{4} & \left( {{USL} < \mu \leq {URT}} \right) \\p_{5} & \left( {{URT} < \mu} \right)\end{matrix}{where}},} \right.} & (44) \\{p_{1} = {Allowance}_{L}} & (45) \\{p_{2} = {p_{1} + p_{{ReworkSuccess}_{L}}}} & (46) \\{p_{3} = 1} & (47) \\{p_{4} = {p_{5} + p_{{ReworkSuccess}_{U}}}} & (48) \\{p_{5} = {Allowance}_{U}} & (49)\end{matrix}$

A filling process in which a machine dispenses product of some kind intoa can is an exemplary process that can be analyzed using methods andtechniques disclosed herein. A measurement used to monitor this processis the Fill Weight of the can. This variable refers to the weight of theproduct in the can after the can has been filled.

For this example, the lower specification limit (LSL) for a can's fillweight has been established as 3.5 pounds. When the weight falls belowthis, the can is opened and 75% of the product is recovered. Theremainder is hauled away by a hazardous waste disposal vendor at a cost.The upper specification limit (USL) for a can's fill weight has beenestablished as 5.5 pounds. If a can exceeds 6.2 pounds, it is deemedineligible for rework and set aside. If a can exceeds 5.5 pounds, butnot 6.2 pounds, it is reworked. Reworking such a can involves removingand discarding some of its product. Following the rework, the can'scontents are again weighed and if these are still found to be above 5.5pounds, it is reworked a second time. If after 4 rework attempts thefill weight is still not within the spec limits, the can is set asidealong with those that were deemed ineligible for rework. Moreover, forthis example, each rework attempt has a 5% chance of being successful(meaning that the fill weight of the can will be reduced to be withinthe spec limits) and a 20% chance of rendering the can so that it cannotbe further reworked (for example, by removing too much product). Perpolicy, of all the cans that are overfilled, regardless of whether ornot they exceed 6.2 pounds, 10% may be shipped as if they were good(i.e. as if they have a weight that is within the spec limits). This 10%is taken from the group of cans which are set aside. The remainder ofthis group is then tagged to be sold at a reduced price.

The costs related to this exemplary process are as follows:

1. The act of filling costs $1.00 per can.

2. The cost of the product with which a can is filled is $0.40 perpound.

3. The hazardous waste disposal fee is $1.50 per can.

4. Reworking a can costs $0.25 per attempt.

5. Unshipped, overfilled cans are sold for $1.25 each.

FIGS. 26A and 26B are schematic views of an exemplary cost model view900 receiving variable inputs. The LSL violation pane 1100 allows theuser to assign lower specification limit properties for each set ofvariable data 122. In the example shown in FIG. 26A, the user can assigna percent of lower specification limit violations allowed 1130 in aninitial disposition pane 1131. A rework radio button 1140 allows theuser to assign whether a part is reworked. A rework settings pane 1152allows the user to assign associated lower specification limit violationrework cost factors 113 d. The rework settings pane 1152 includes amaximum rework attempts radio button 1153 for assigning whether toenforce a maximum number of rework attempts 1154. A rework thresholdradio button 1155 allows the user to assign whether to enforce a reworkthreshold 1156 a, a cost of each attempt 1156 b, a chance of success1156 c, and a chance of scrap 1156 d. A final disposition pane 1158allows the user to assign associated lower specification limit violationscrap cost factors 113 c and, in the example shown, includes a disposalcost 1158 b, a scrap recovery cost 1158 c, and a percent materialrecovery 1158 d.

The USL violation pane 1200 allows the user to assign upperspecification limit properties for each set of variable data 122. In theexample shown in FIG. 26A, the user can assign a percent of upperspecification limit violations allowed 1230 in an initial dispositionpane 1231. A rework radio button 1240 allows the user to assign whethera part is reworked. A rework settings pane 1252 allows the user toassign associated upper specification limit violation rework costfactors 113 f. The rework settings pane 1252 includes a maximum reworkattempts radio button 1253 for assigning whether to enforce a maximumnumber of rework attempts 1254. A rework threshold radio button 1255allows the user to assign whether to enforce a rework threshold 1256 a,a cost of each attempt 1256 b, a chance of success 1256 c, and a chanceof scrap 1256 d. A final disposition pane 1258 allows the user to assignassociated upper specification limit violation scrap cost factors 113 eand, in the example shown, includes a disposal cost 1258 b, a scraprecovery cost 1258 c, and a percent material recovery 1258 d.

The cost inspector cross-correlates the received inputs with acollection of production data. FIG. 26C provides an output view 2600 ofthe cost inspector providing a graphical representation of thecross-correlation. FIG. 26D provides a detail output view 2602 of thecost inspector providing a graphical representation of thecross-correlation.

Following a similar thought process to that used in the earliermachining case study, each cost factor will be found to contribute tothe manufacturing costs of a single unit, or the percentage chance thata manufactured unit is good (i.e., saleable), or both. To start theanalysis, consider a single unit manufactured between the specificationlimits. The total manufacturing cost for each such unit is the FillWeight times the Per Unit Of Measure cost (e.g., 4.2 lbs.*0.40$/lb.=$1.68) plus the filling cost (in this example, $1.00 per can).This is the equation of a line given as:

C3(x)=0.40*x+1.00(LSL<x<USL)  (50)

Further, 100% of the units manufactured having a Fill Weight between thespecification limits are (by definition) good and saleable. Therefore,

M3(x)=1.00(LSL<x<USL)  (51)

To arrive at the proper model equations for those units manufacturedhaving a Fill Weight less than the LSL, we must adjust the costs justcalculated by 3 factors. First, none of the units manufactured below theLSL are good, therefore,

M1(x)=0.00(x<LSL)  (52)

Second, the cost of each such unit is reduced by 0.75*0.40*x, since 75%of the product put in the can is recovered for future use. And finally,a hazardous waste disposal fee of $1.50 must be added to the cost tomanufacture such a unit. Putting these together gives,

C1(x)=(1.00−0.75)*0.40*x+1.00+1.50(x<LSL)  (53)

Turning our attention to those units which are initially overfilled, wehave two possible process paths to pursue. First, in this example, thoseunits with a Fill Weight of more than 6.2 lbs. are not eligible forre-work, but 10% of these units can be sold for full price. The other90% can be sold for $1.25 per unit (presumably, a fraction of what agood unit is sold for.) This leads to the following relationships,

$\begin{matrix}\begin{matrix}{{C\; 5(x)} = {{0.10\left( {{0.40*x} + 1.00} \right)} + {0.90\left( {{0.40*x} + 1.00 - 1.25} \right)}}} \\{\left( {6.20 < x} \right)} \\{= {{0.40*x} - 0.125}}\end{matrix} & (54) \\{{M\; 5(x)} = {0.10\left( {6.20 < x} \right)}} & (55)\end{matrix}$

The remaining group of overfilled units consists of those units whoseinitial Fill Weight falls between the USL and 6.20 lbs. Each of theseunits will be reworked up to 4 times at a cost of $0.25 per reworkattempt. With each attempt, we have a 5% chance of successfullyadjusting the Fill Weight such that it falls between the specificationlimits, and a 20% chance of damaging the unit such that no furtherrework attempts can be made. This implies that we have a 75% chance ofrequiring another rework attempt. Using standard methods, we cancalculate the chance of successfully reworking a unit (in up to 4attempts) as,

$\begin{matrix}\begin{matrix}{p_{Success} = {\left( {5\%} \right) + \left( {75\%*5\%} \right) + \left( {75\%*75\%*5\%} \right) +}} \\{\left( {75\%*75\%*75\%*5\%} \right)} \\{= {0.05*\left( {1 + 0.75 + {0.75*0.75} + {0.75*0.75*0.75}} \right)}} \\{= {13.672\%}}\end{matrix} & (56)\end{matrix}$

Adding this percentage to the 10% of units in this range that we areallowed to ship means that,

M4(x)=0.23672(USL<x<6.20)  (57)

When attempting to calculate the manufacturing costs accrued by a unitin this range of Fill Weights, we must first calculate the cost ofmanufacturing a successfully reworked unit, then the cost of a unitwhich failed its rework. Finally, these two costs are combined based onthe percentage just calculated. That is, 13.672% of units manufacturedwill have the “good” cost, and 86.328% will have the “bad” cost.

The “good” cost is simply C3 (above) plus the average cost of asuccessful rework. Using standard methods, the cost of a successfulrework is,

c_(Success)=(5%*$0.25)+(75%*5%*2*$0.25)+(75%*75%*5%*3*$0.25)+(75%*75%*75%*5%*4*$0.25)  (58)

That is, 5% of the units reworked, will only be reworked one time at acost of $0.25. 5% of the 75% of the units that survive the first reworkattempt, will be successfully reworked on the second attempt at a totalcost of $0.50. And so on, up to 4 attempts. In total, then,

c _(Success)=$0.07344 per unit (on average)  (59)

While it may seem odd that the average cost of a successful rework isless than the cost of any given rework attempt, this number also factorsin the chance of successfully reworking a unit.

In a similar manner, the cost of a failed rework is,

c_(Failure)=(20%*$0.25)+(75%*20%*2*$0.25)+(75%*75%*20%*3*$0.25)+(75%*75%*75%*95%*4*$0.25)  (60)

or,

c _(Failure)=$0.61016 per unit (on average)  (61)

Putting this altogether, then, and including the $1.25 salvage value forthose units failing rework,

C4(x)=0.40*x+13.672%*(1.00+0.07344)+86.328%*(−0.125+0.61016)(USL<x<6.20)  (62)

or, more simply,

C4(x)=0.40*x+0.5656(USL<x<6.20)  (63)

We have now fully developed the M_(good)(x) and C_(good)(x) equationsnecessary for the final analysis. Restated here for clarity, we have,

$\begin{matrix}{{M_{good}(x)} = \left\{ {{\begin{matrix}{M_{1} = 0} & \left( {x < {LSL}} \right) \\{M_{3} = 1} & \left( {{LSL} < x < {USL}} \right) \\{M_{4} = 0.23672} & \left( {{USL} < x < 6.20} \right) \\{M_{5} = 0.1} & \left( {6.20 < x} \right)\end{matrix}{and}},} \right.} & (64) \\{{C_{good}(x)} = \left\{ \begin{matrix}{C_{1} = {{0.10*x} + 2.50}} & \left( {x < {LSL}} \right) \\{C_{3} = {{0.40*x} + 1.00}} & \left( {{LSL} < x < {USL}} \right) \\{C_{4} = {{0.40*x} + 0.5656}} & \left( {{USL} < x < 6.20} \right) \\{C_{5} = {{0.40*x} - 0.125}} & \left( {6.20 < x} \right)\end{matrix} \right.} & (65)\end{matrix}$

The first of the above two calculations results in a piecewise constantcurve. The second results in a piecewise linear curve.

As in the first use case, to get to the average, effective cost perunit, we must cross-correlate each of these functions with theprobability density function of the manufacturing data we are analyzing,and then divide the results. Symbolically,

$\begin{matrix}{{{AEC}\left( {\mu_{0} + \delta} \right)} = {\frac{R_{xy}(\delta)}{R_{xz}(\delta)} = \frac{\int_{- \infty}^{\infty}{{{C_{good}\left( {\zeta + \delta} \right)} \cdot p}{{f(\zeta)}}{\zeta}}}{\int_{- \infty}^{\infty}{{{M_{good}\left( {\zeta + \delta} \right)} \cdot p}{{f(\zeta)}}{\zeta}}}}} & (66)\end{matrix}$

As before, the minimum value of this function (the minimum averageeffective per-unit manufacturing cost) can be calculated using standardmethods.

Various implementations of the systems and techniques described here canbe realized in digital electronic circuitry, integrated circuitry,specially designed ASICs (application specific integrated circuits),computer hardware, firmware, software, and/or combinations thereof.These various implementations can include implementation in one or morecomputer programs that are executable and/or interpretable on aprogrammable system including at least one programmable processor, whichmay be special or general purpose, coupled to receive data andinstructions from, and to transmit data and instructions to, a storagesystem, at least one input device, and at least one output device.

These computer programs (also known as programs, software, softwareapplications or code) include machine instructions for a programmableprocessor, and can be implemented in a high-level procedural and/orobject-oriented programming language, and/or in assembly/machinelanguage. As used herein, the terms “machine-readable medium” and“computer-readable medium” refer to any computer program product,apparatus and/or device (e.g., magnetic discs, optical disks, memory,Programmable Logic Devices (PLDs)) used to provide machine instructionsand/or data to a programmable processor, including a machine-readablemedium that receives machine instructions as a machine-readable signal.The term “machine-readable signal” refers to any signal used to providemachine instructions and/or data to a programmable processor.

Implementations of the subject matter and the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, or in computer software, firmware, or hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. Embodiments ofthe subject matter described in this specification can be implemented asone or more computer program products, i.e., one or more modules ofcomputer program instructions encoded on a computer readable medium forexecution by, or to control the operation of, data processing apparatus.The computer readable medium can be a machine-readable storage device, amachine-readable storage substrate, a memory device, a composition ofmatter effecting a machine-readable propagated signal, or a combinationof one or more of them. The term “data processing apparatus” encompassesall apparatus, devices, and machines for processing data, including byway of example a programmable processor, a computer, or multipleprocessors or computers. The apparatus can include, in addition tohardware, code that creates an execution environment for the computerprogram in question, e.g., code that constitutes processor firmware, aprotocol stack, a database management system, an operating system, or acombination of one or more of them. A propagated signal is anartificially generated signal, e.g., a machine-generated electrical,optical, or electromagnetic signal, that is generated to encodeinformation for transmission to suitable receiver apparatus.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a stand alone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Moreover, a computer can be embedded inanother device, e.g., a mobile telephone, a personal digital assistant(PDA), a mobile audio player, a Global Positioning System (GPS)receiver, to name just a few. Computer readable media suitable forstoring computer program instructions and data include all forms of nonvolatile memory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto optical disks; and CD ROM and DVD-ROM disks. The processor andthe memory can be supplemented by, or incorporated in, special purposelogic circuitry.

Implementations of the subject matter described in this specificationcan be implemented in a computing system that includes a back endcomponent, e.g., as a data server, or that includes a middlewarecomponent, e.g., an application server, or that includes a front endcomponent, e.g., a client computer having a graphical user interface ora Web browser through which a user can interact with an implementationof the subject matter described is this specification, or anycombination of one or more such back end, middleware, or front endcomponents. The components of the system can be interconnected by anyform or medium of digital data communication, e.g., a communicationnetwork. Examples of communication networks include a local area network(“LAN”) and a wide area network (“WAN”), e.g., the Internet.

The computing system can include clients and servers. A client andserver are generally remote from each other and typically interactthrough a communication network. The relationship of client and serverarises by virtue of computer programs running on the respectivecomputers and having a client-server relationship to each other.

While this specification contains many specifics, these should not beconstrued as limitations on the scope of the invention or of what may beclaimed, but rather as descriptions of features specific to particularimplementations of the invention. Certain features that are described inthis specification in the context of separate implementations can alsobe implemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multi-tasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the embodiments described above should not be understoodas requiring such separation in all embodiments, and it should beunderstood that the described program components and systems cangenerally be integrated together in a single software product orpackaged into multiple software products.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made without departingfrom the spirit and scope of the disclosure. Accordingly, otherimplementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results.

1. A method of process cost analysis, the method comprising: determininga per-unit cost function for executing a process step; determining apercentage-of-acceptable-parts function for executing a process step;receiving production data into memory, the production data correspondingto a measured quality metric of the executed process step; determining aprobability density function for the received production data; executingon a processor a correlation routine for cross-correlating the per-unitcost function with the probability density function of the productiondata to provide a first cross-correlation; executing on the processorthe correlation routine for cross-correlating thepercentage-of-acceptable-parts function with the probability densityfunction of the production data to provide a second cross-correlation;determining an average effective per-unit cost to produce a resultant ofthe process step by dividing the first cross-correlation by the secondcross-correlation.
 2. The method of claim 1, further comprisingdetermining a minimum value of the quotient of the first and secondcross-correlations to determine a minimum average effective per-unitcost associated with a current data distribution.
 3. The method of claim2, further comprising electronically displaying a plot corresponding tothe determined minimum average effective per-unit cost.
 4. The method ofclaim 2, further comprising determining an optimal process meanassociated with the minimum average effective per-unit cost associatedwith a current data distribution.
 5. The method of claim 4, furthercomprising presenting a process offset based at least in part on adifference between the optimal process mean and a current process meanassociated with a current data distribution, the process offsetassociated with a cost savings of the process step.
 6. The method ofclaim 2, further comprising iteratively determining the minimum averageeffective per-unit cost using different standard deviations for theprobability density function of the received data corresponding to atheoretical reduction in process variability.
 7. The method of claim 6,further comprising determining an optimal mean and minimum averageeffective per-unit cost at one or more standard deviations.
 8. Themethod of claim 6, further comprising presenting the iterativelydetermined minimum average effective per-unit costs.
 9. A method ofprocess cost analysis, the method comprising: receiving at least onecost factor into memory for executing a process step; receivingproduction data into memory, the production data corresponding to ameasured quality metric of the executed process step; determining a perunit manufacturing cost for the process step; determining a percentageof acceptable units manufactured by the process step; modeling the perunit manufacturing cost as a piecewise linear function; modeling thepercentage of acceptable units as a piecewise constant function;determining a probability density function for the received productiondata; determining a first cross-correlation by executing on a processora correlation routine for cross-correlating the piecewise linearfunction with the probability density function for the receivedproduction data; determining a second cross-correlation by executing onthe process the cross-correlation routine for cross-correlating thepiecewise constant function with the probability density function forthe received production data; and determining an average effectiveper-unit cost to produce a resultant of the process step by dividing thefirst cross-correlation by the second cross-correlation.
 10. The methodof claim 9, further comprising determining a minimum value of thequotient of the first and second cross-correlations to determine aminimum average effective per-unit cost associated with a current datadistribution.
 11. The method of claim 9, further comprisingelectronically displaying a plot corresponding to the determined averageeffective per-unit cost.
 12. The method of claim 9, further comprisingdetermining an optimal process mean associated with the minimum averageeffective per-unit cost associated with a current data distribution. 13.The method of claim 12, further comprising presenting a process offsetbased at least in part on a difference between the optimal process meanand a current process mean associated with a current data distribution,the process offset associated with a cost savings of the process step.14. The method of claim 9, further comprising iteratively determiningthe minimum average effective per-unit cost using different standarddeviations for the probability density function of the received datacorresponding to a theoretical reduction in process variability.
 15. Themethod of claim 14, further comprising determining an optimal mean andminimum average effective per-unit cost at one or more standarddeviations.
 16. The method of claim 14, further comprising presentingthe iteratively determined minimum average effective per-unit costs. 17.The method of claim 9, wherein the at least one cost factor comprisingat least one of: a lower specification limit violation scrap costfactor; a lower specification limit violation rework cost factor; anupper specification limit violation scrap cost factor; an upperspecification limit violation rework cost factor; a maximum reworkattempts cost factor for lower specification limit violations; a maximumrework attempts cost factor for upper specification limit violation; alower dimensional limit for rework threshold cost factor; an upperdimensional limit for rework threshold cost factor; a cost of eachrework attempt cost factor for lower specification limit violations acost of each rework attempt cost factor for upper specification limitviolations a percentage chance of rework success cost factor for lowerspecification limit violations; a percentage chance of rework successcost factor for upper specification limit violations; a percentagechance of rework failure cost factor for lower specification limitviolations; a percentage chance of rework failure cost factor for upperspecification limit violations; a percentage of lower specificationlimit violations allowed cost factor a percentage of upper specificationlimit violations allowed cost factor a disposal cost of a scrapped unitcost factor for lower specification limit violations; a disposal cost ofa scrapped unit cost factor for upper specification limit violations; aper unit recovery value of a scrapped unit cost factor for lowerspecification limit violations; a per unit recovery value of a scrappedunit cost factor for upper specification limit violations; a per unitmaterial value of a scrapped unit cost factor for lower specificationlimit violations; a per unit material value of a scrapped unit costfactor for upper specification limit violations; a per step process costfactor; and a per unit of measure material cost factor.
 18. A costinspection system for a process, the system comprising: a functionmodule receiving a cost model associated with the at least one processstep and providing a per-unit cost function and apercentage-of-acceptable-parts function for executing a process step; adata module receiving production data corresponding to a measuredquality metric of the executed process step and providing a probabilitydensity function for the received production data; a cross-correlationmodule cross-correlating the per-unit cost function with the probabilitydensity function of the production data to provide a firstcross-correlation and cross-correlating thepercentage-of-acceptable-parts function with the probability densityfunction of the production data to provide a second cross-correlation;and a cost module executing on a computing processor and determining anaverage effective per-unit cost to produce a resultant of the at leastone process step by dividing the first cross-correlation by the secondcross-correlation.
 19. The system of claim 18, wherein the cost moduledisplays on an electronic device a plot corresponding to the determinedaverage effective per-unit cost.
 20. The system of claim 18, wherein thecost module determines a minimum value of the quotient of the first andsecond cross-correlations to determine a minimum average effectiveper-unit cost associated with a current data distribution.
 21. Thesystem of claim 20, wherein the cost module determines an optimalprocess mean associated with the minimum average effective per-unit costassociated with a current data distribution.
 22. The system of claim 21,wherein the cost module presents a process offset based at least in parton a difference between the optimal process mean and a current processmean associated with a current data distribution, the process offsetassociated with a cost savings of the at least one process step.
 23. Thesystem of claim 20, wherein the cost module iteratively determines theminimum average effective per-unit cost using different standarddeviations for the probability density function of the received datacorresponding to a theoretical reduction in process variability.
 24. Thesystem of claim 23, wherein the cost module determines an optimal meanand a minimum average effective per-unit cost at one or more standarddeviations.
 25. The system of claim 24, wherein the cost moduleelectronically displays the iteratively determined minimum averageeffective per-unit costs.
 26. The system of claim 18, wherein the costmodule receives at least one cost factor comprising at least one of: alower specification limit violation scrap cost factor; a lowerspecification limit violation rework cost factor; an upper specificationlimit violation scrap cost factor; an upper specification limitviolation rework cost factor; a maximum rework attempts cost factor forlower specification limit violations; a maximum rework attempts costfactor for upper specification limit violation; a lower dimensionallimit for rework threshold cost factor; an upper dimensional limit forrework threshold cost factor; a cost of each rework attempt cost factorfor lower specification limit violations a cost of each rework attemptcost factor for upper specification limit violations a percentage chanceof rework success cost factor for lower specification limit violations;a percentage chance of rework success cost factor for upperspecification limit violations; a percentage chance of rework failurecost factor for lower specification limit violations; a percentagechance of rework failure cost factor for upper specification limitviolations; a percentage of lower specification limit violations allowedcost factor a percentage of upper specification limit violations allowedcost factor a disposal cost of a scrapped unit cost factor for lowerspecification limit violations; a disposal cost of a scrapped unit costfactor for upper specification limit violations; a per unit recoveryvalue of a scrapped unit cost factor for lower specification limitviolations; a per unit recovery value of a scrapped unit cost factor forupper specification limit violations; a per unit material value of ascrapped unit cost factor for lower specification limit violations; aper unit material value of a scrapped unit cost factor for upperspecification limit violations; a per step process cost factor; and aper unit of measure material cost factor.